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Given A = 2 & 1 5 & 3 , we have a=2, b=1, c=5, and d=3.

May 6, 2026Mathematics

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Answer

\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}

3 steps

To find the inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the formula is:

A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Identify the elements of matrix $A$ .
Given $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$ , we have $a=2$ , $b=1$ , $c=5$ , and $d=3$ .
Calculate the determinant of $A$ .
The determinant is $ad - bc$ .
$\det(A) = (2)(3) - (1)(5) = 6 - 5 = 1$
Apply the inverse formula.
Substitute the values into the formula.
$A^{-1} = \frac{1}{1} \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$ $A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$
Comparing this result with the given options, the correct answer is the fourth option.

The final answer is $\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$ .

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